(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other.

(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other.